The problem is asking for the derivative $\frac{d}{dx} f(g(x))$ at $x = 6$, using the given table of values.

This requires the application of the chain rule, which states that:

$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

We need to evaluate this at $x = 6$.

Step-by-step solution:

1. From the table, when $x = 6$, we have:

   • $g(6) = 6$
   • $g'(6) = 5$

2. Now, we need $f'(g(6))$. Since $g(6) = 6$, we find $f'(6)$ from the table:

   • $f'(6) = 6$

3. Applying the chain rule: $\frac{d}{dx} f(g(x))\bigg|_{x=6} = f'(g(6)) \cdot g'(6) = f'(6) \cdot g'(6) = 6 \cdot 5 = 30$

Thus, the derivative $\frac{d}{dx} f(g(x))$ at $x = 6$ is 30.

Would you like further details or have any other questions?

Here are 5 related questions:

1. What is the chain rule in calculus, and how is it applied?
2. How would the solution change if the function $g(x)$ had different values at $x = 6$?
3. Can you compute $\frac{d}{dx} g(f(x))$ at $x = 6$ using the same table?
4. How would you find the second derivative of $f(g(x))$ at $x = 6$?
5. How do the values of $f'(x)$ and $g'(x)$ impact the rate of change of the composition $f(g(x))$?

Tip: Always remember to apply the chain rule when dealing with composite functions!